Re: Logarithm of transfinite numbers

stephen@xxxxxxxxxx wrote:
matt271829-news@xxxxxxxxxxx wrote:

Randy Poe wrote:
matt271829-news@xxxxxxxxxxx wrote:

It doesn't seem right because it isn't right. The bit string
representations of natural numbers, whatever representation
you choose, consist of only a countable subset of the
uncountable bit strings of length aleph_0.
For instance
you might choose to represent the natural numbers as
...001, ...010, ...100, ...1000, i.e, infinite bit strings where only
a single bit is nonzero.

Yes, we *can* represent them this way, but we can already list them as
1,2,3,4... which we know is countable. You're just putting one bit in
the positions 1,2,3,4... so what new does this tell us?

Imagine instead that we *do* represent them in proper binary form. We
still need aleph_0 bits, and every arrangement of those aleph_0 bits
seems to map to a unique natural number.

No, every arrangement of those aleph_0 bits does not seem to map
to a unique natural number. Which natural number is mapped to
..1111111111111
or
..0101010101010


I assume ".." means a (countably) infinite series. They would both have
to be non-finite, I suppose, if that's what you mean. Are you saying,
very roughly speaking, that the problem is that the "number infinity"
appears infinitely many times in all those arrangements of aleph_0
bits?

Only the bit strings which have an infinite number of leading
zero's correspond to a natural number, and there are only aleph_0
such bit strings.

This is where it goes wrong,
and where the flaw in the logic must lie. It seems to me that we need
to somehow show that this one-to-one mapping can't be made.

That is what Cantor did all those years ago.

Didn't he go bonkers or something? I can't say I'm surprised!

.

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